Base unit (measurement)

In the International System of Units, there are seven base units: kilogram, metre, candela, second, ampere, kelvin, and mole.

A set of fundamental dimensions of physical quantity is a minimal set of units such that every physical quantity can be expressed in terms of this set. The traditional fundamental dimensions of physical quantity are mass, length, time, charge, and temperature, but in principle, other fundamental quantities could be used. Electric current could be used instead of charge or speed could be used instead of length. Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy (or mass, length, and time). In addition, some physicists recognize electric charge as a separate fundamental dimension of physical quantity, even if it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. There are also physicists who have cast doubt on the very existence of incompatible fundamental quantities[1].

There are other relationships between physical quantities that can be expressed by means of fundamental constants, and to some extent it is an arbitrary decision whether to retain the fundamental constant as a quantity with dimensions or simply to define it as unity or a fixed dimensionless number, and reduce the number of explicit fundamental constants by one. The ontological issue is whether these fundamental constants really exist as dimensional or dimensionless quantities. This is equivalent to treating length as the same commensurable physical material as time or understanding electric charge as a combination of quantities of mass, length, and time which may seem less natural than thinking of temperature as measuring the same material as energy (which is expressible in terms of mass, length, and time).

For instance, time and distance are related to each other by the speed of light, c, which is a fundamental constant. It is possible to use this relationship to eliminate either the unit of time or that of distance. Similar considerations apply to the Planck constant, h, which relates energy (with dimension expressible in terms of mass, length and time) to frequency (with dimension expressible in terms of time). In theoretical physics it is customary to use such units (natural units) in which c = 1 and ħ = 1. A similar choice can be applied to the vacuum permittivity, ε₀.

• One could eliminate either the metre or the second by setting c to unity (or to any other fixed dimensionless number).
• One could then eliminate the kilogram by setting ħ to a dimensionless number.
• One could then further eliminate the ampere by setting either the vacuum permittivity ε₀ (alternatively, the Coulomb constant k_e = 1/(4πε₀)) or the elementary charge e to a dimensionless number.
• One could eliminate the mole as a base unit by setting the Avogadro constant N_A to 1. This is natural as it is a technical scaling constant.
• One could eliminate the kelvin as it can be argued that temperature simply expresses the energy per particle per degree of freedom, which can be expressed in terms of energy (or mass, length, and time). Another way of saying this is that Boltzmann's constant k_B is a technical scaling constant and could be set to a fixed dimensionless number.
• Similarly, one could eliminate the candela, as that is defined in terms of other physical quantities via a technical scaling constant, K_cd.
• That leaves one base dimension and an associated base unit, but there are several fundamental constants left to eliminate that too – for instance, one could use G, the gravitational constant, m_e, the electron rest mass, or Λ, the cosmological constant.

A widely used choice, in particular for theoretical physics, is given by the system of Planck units, which are defined by setting ħ = c = G = k_B = k_e = 1.

Using natural units leaves every physical quantity expressed as a dimensionless number, which is noted by physicists disputing the existence of incompatible fundamental physical quantities.[1][2][3]